Date: 4/5/96 at 11:24:1
From: Doctor Aaron
Subject: Re: Need help please
Hi Jodi,
Working with fractions of algebraic expressions is just like
working with fractions of integers. One thing we have to do is
find a common denominator. Since you were asked to express the
result in lowest terms, we also have to worry about that.
In problem 1 you have a product of two expressions. If it were
5 6
- * -
3 15
you'd want to do some canceling, but with algebraic expressions
it's a little harder to see what cancels with what and what
will be left over.
With algebraic expressions we just have to define what we mean by
a factor.
Well 6xy isn't too bad - it's just 6*x*y so the factors are 6, x
and y.
Now we have to figure out x^2-y^2. Well, we can rewrite this as
x^2 - xy + xy - y^2, which may seem silly at first, but it will be
useful. Then we can factor out an x from the first two pieces to
rewrite it again as:
x*(x-y) + xy - y^2
We can do the same for y and the last two pieces to get
x*(x-y) + y*(x-y), which we can rewrite as:
(x+y)*(x-y).
Then the factors, or multiples, are (x+y) and (x-y)
This is the process known as factoring. We have broken up a
sum of variables to the second power to a product of sums of
variables to the first power.
Try to work through x^2 - 2xy - y^2 by yourself. You should get
(x-y)*(x-y).
Once you believe that x^2-2xy-y^2 = (x-y)*(x-y), we can rewrite
your equation as:
(x-y)*(x-y) 3*x^2*y^2
---------- * -----------
6*x*y (x+y)*(x-y)
we can cancel an x-y to get
(x-y) 3*x^2*y^2
----- * ---------
6*x*y (x+y)
but we can still simplify because we can write x^2 as x*x and y^2
as y*y and then cancel to get:
(x-y) 3*x*y (x-y)*x*y
----- * ---- but 3*2 = 6 so we can rewrite again: ---------
6 (x+y) 2*(x+y)
The second problem is a little different. Instead of a product,
you have to calculate a difference. Luckily you already have a
common denominator, x-1. You can multiply both sides of the
equation by x-1 just as you would multiply by 3 if you saw:
4 + 5 - x 22
- - - = -
3 1 3 3
After you multiply both sides by x-1, the denominators will cancel
and you'll be left with 5 - 4x = 2 - x.
Good luck.
-Doctor Aaron, The Math Forum